In the oil and gas industry, seismic prospecting techniques are commonly used to aid in the search for and evaluation of subterranean hydrocarbon deposits. In seismic prospecting, a seismic source is used to generate a seismic signal which propagates into the earth and is at least partially reflected by subsurface seismic reflectors (i.e., interfaces between underground formations having different elastic properties). The reflections are detected and measured by seismic receivers located at or near the surface of the earth, in an overlying body of water, or at known depths in boreholes, and the resulting seismic data may be processed to yield information relating to the subsurface formations.
Seismic prospecting consists of three main stages: data acquisition, data processing, and interpretation. The success of a seismic prospecting operation is dependent on satisfactory completion of all three stages.
The most widely used seismic data acquisition and processing technique is the common midpoint (CMP) method. The midpoint is the point midway between the source and the receiver. According to the CMP method, each seismic signal is recorded at a number of different receiver locations and each receiver location is used to record seismic signals from a number of different source locations. This results in a number of different data traces having different source-to-receiver offsets for each midpoint. The resulting data traces are corrected for normal moveout (i.e., the variation of reflection arrival time caused by variation of the source-to-receiver offset), and are then sorted into common midpoint gathers and stacked to simulate the data trace that would have been recorded by a coincident source and receiver at each midpoint location, but with improved signal-to-noise ratio.
The primary purposes of seismic data processing are to remove or suppress unwanted noise components and to transform the data into seismic sections or images which facilitate interpretation. Examples of well known seismic data processing operations include applying corrections for known perturbing causes, rearranging the data, filtering the data according to a selected criteria, stacking the data, migrating the stacked data to correctly position the reflectors, measuring attributes of the data, and displaying the final result. The particular sequence of processing operations used to process seismic data is dependent on a number of factors such as the field acquisition parameters, the quality of the data, and the desired output.
As is well known in the art, modern seismic data processing operations are performed on digitized data. A digitized seismic data trace is a uniformly sampled time series of discrete measurements of the seismic signal at the receiver in question. A problem known as "aliasing," which is defined as the introduction of frequency ambiguities as a result of the sampling process, can occur in processing digitized data. Where there are fewer than two data samples per cycle, an input signal at one frequency can appear to be another frequency at output. Hence, this problem is best described in the frequency domain where aliased frequencies can be folded (or wrapped) onto other frequencies. To avoid aliasing, frequencies above the folding or Nyquist frequency (which is one-half of the frequency of sampling) must be removed by an anti-alias filter before sampling. For example, for seismic data having a 4 millisecond sampling rate (i.e., a sampling frequency of 250 samples per second), all frequencies above 125 Hertz must be removed in order to avoid aliasing.
Aliasing is an inherent property of all sampling systems. It applies to sampling at discrete time intervals in digital seismic recording, as described above. It can also apply to the spatial distance between individual CMP locations (i.e., spatial sampling). If the spatial sampling interval is too large, certain data processing operations performed in the spatial frequency domain (e.g., dip moveout or DMO) can become spatially aliased. If this happens, events with steep dips can be perceived as different from what they actually are and acquisition noises can be introduced into the processed data.
The problem of spatial aliasing can be particularly acute with respect to a three-dimensional (3-D) marine seismic survey in which multiple sources and/or multiple receivers are used to generate multiple CMP lines on each pass of the seismic vessel resulting in fewer passes (and, consequently, lower cost) to complete the survey. Each of these CMP lines, however, typically has a number of missing shots relative to normal two-dimensional (2-D) data acquisition. These missing shots can cause spatial aliasing during subsequent data processing operations which will degrade the final image.
Spatial aliasing can be avoided by using a sufficiently small trace spacing. This requires either (a) modification of the field recording geometry to include additional sources and/or receivers or (b) use of a data-dependent interpolation scheme to generate extra traces.
Modification of the field recording geometry to generate additional data traces is undesirable because of the added expense that this entails. Therefore, past efforts to avoid spatial aliasing have focused on statistical interpolation techniques to generate additional data traces. One such technique is FX-Wiener interpolation. See, e.g., Spitz, S., "Seismic trace interpolation in the F-X domain," Geophysics, Vol. 56, No. 6, June 1991, pp. 785-794. However, this technique has proven to be too unstable for routine use. It is based on least squares matching of the data using a spatially invariant filter. The least squares criterion tends to ignore primary data when a higher amplitude coherent noise is present. The spatial invariance of the filter favors constant dipping reflectors over curved ones, since only constant dipping reflectors are truly predictable using a spatially invariant filter. Also, extrapolation of an event into a gap using FX-Wiener interpolation can produce very high amplitudes if the amplitude of the event is increasing in the direction of the gap.
Many of the flaws of FX-Wiener interpolation are also present in any statistical interpolation technique. The source of the difficulties is relying entirely on the recorded data to predict the missing data. Recorded data are contaminated with coherent and incoherent noises which make signal interpolation or extrapolation difficult.
Another potential method for solving the aliasing problem is based on the work of Vermeer. This method assumes that all data in a common midpoint gather have positive moveout. Hence, in k-space, any data which appear at negative k values were wrapped around from the positive values. If the data are only aliased once, the negative k values can be mapped to their corresponding positive values, effectively halving the spacing in the CMP domain and doubling the aliasing frequency. This approach works well so long as the data are aliased, or wrapped, no more than once. However, this method cannot be used to unwrap data which are aliased (wrapped) more than once.
What is needed is a deterministic interpolation technique which uses knowledge about the underlying wave-propagation theory to interpolate or extrapolate data. The present invention satisfies this need.